Aleph One wrote "Smashing The Stack For Fun And Profit." [1] I found this Aleph Null through a citation in "The UNIX Time-Sharing System" by Dennis Ritchie and Ken Thompson [2]
In case anyone doesn't know, Aleph-null is the cardinality of the natural numbers and Aleph-one is the cardinality of the countable ordinal numbers. On the continuum hypothesis, aleph-one would equal the cardinality of the continuum (i.e. reals.)
I think you have a mistake in your aleph-one summary, if I recall correctly aleph-zero is the cardinality of countably infinite set (like natural numbers), and aleph-one is for uncountably infinite sets. It's been a while though so I may be misremembering.
Unless guerrilla's comment was edited and you're referring to an older version, guerrilla's comment is completely correct.
Aleph_1 doesn't refer to uncountable cardinals in general, it's a specific uncountable cardinal. More specifically, it is, as guerrilla says, the number of countable ordinals.
Aleph_1 refers to the second-smallest infinite cardinal, whether that's equal to 2^(aleph_0) or not. (Aleph_1 is also equal, as has been mentioned, to the number of countable ordinals.)
The continuum hypothesis, in stating that there are no infinite cardinals inbetween aleph_0 and 2^(aleph_0), is equivalent to the statement that 2^(aleph_0) = aleph_1.
The series you refer to is denoted by the Hebrew letter bet, not aleph. So bet_0 = aleph_0, bet_1 = 2^(bet_0), bet_2 = 2^(bet_1), etc. But that is not how the aleph numbers work.
Some of us prefer to assume the continuum hypothesis just for the sake of this notation... but it turns out that the continuum hypothesis is still pretty hot.
"Assuming the continuum hypothesis for the sake of this notation" is being deliberately obfuscatory, encouraging confusion, and requiring other mathematicians to do extra work to translate your results to the more general context. There's perfectly good notation for the bet series, and no need to use aleph as a substitute when that simply isn't what it means.
I am also rusty, but I believe both statements are true but mine may be more precise as there is more than one uncountable infinite set and the set of all countable ordinal numbers is an uncountable infinite set.
The cardinality of Real numbers is 2^Aleph_Null. Continuum hypothesis is equivalent to saying that it is equal to Aleph_One. But that is debated :) [Debated in the sense that you could accept one or the other, but it is independent of ZF and there should be a good reason to accept one or the other]
